Closed Geodesics on Euclidean Homogeneous Spaces
When you talk to about studying geometry, you often find them recalling memories of the Pythagorean theorem, or cyclic geometry from their high school maths classes. Of course, while this is a form of geometry, it’s far from the modern geometry that is so pervasive in the modern sciences.
Modern geometry is far from the ages of drawing lines on a page with a straight edge and compass. Even so, we still care about the “nice” properties that make that page so useful to the practical sciences. Notions of area, volume, parallelism, and length are all key properties that modern geometry studies. Whilst the paper is now a manifold, and the rulers are complicated functions defined on local neighbourhoods of these structures, the key concepts are still alive.
A key part of my project over the summer was to investigate what we mean when we say a line is “straight”. As simple as this question sounds, there are numerous ways to interpret the word “straight”. This may sound overly semantic, but it’s useful to pin down exactly what we mean when we define something in a new light.
As we’ve come to learn from drawing lines on paper, a straight line is just the line that minimises the distance between two points. Maybe this is the right definition of straight lines. Unfortunately, when we want to look at straight lines in more interesting spaces, for example the sphere, the straight lines between two points can be connected in one of two ways. Either we can take the shortest path around the equator, or we can move the opposite direction and eventually reach the same place. The latter of these lines is certainly not the shortest path, but it is as straight as the first. This second path however is always the shortest path between “close” points on the sphere. It turns out the notion of local distance minimisation is a good definition of a straight line, and it reflects a lot of the physical laws we expect to be able to derive from our flat paper geometry from before.
We may ask ourselves if there are other intuitive ways to define what we mean by straight. Using the Earth as an example, we already know what a straight line is. That is, if we walk without turning where we are facing, then we are moving in a straight line. This generalises back to the flat sheet of paper very nicely, since if we start walking one way, and don’t turn at all, we will draw out a straight line.
It turns out that these two definitions of straight lines are equivalent to each other. In fact, any reasonable definition of a straight line should always be equivalent to these definitions. Throughout the summer I studied a number of these definitions in trying to learn more about the nature of geometry. In doing so, I learned that seemingly obvious things can be extremely difficult to pin down in a good way.
Rohin Berichon was a recipient of a 2018/19 AMSI Vacation Research Scholarship.